The proposed method is valid for a special type of primary exposure such that it can readily be obtained from existing study data repository. When treatment is the primary exposure, treatment assignment status or level of the treatment information given to a par- ticipant can be obtained with relatively less much cost and efforts than expensive covariate information. However, the proposed method is not suitable for studies in the presence of missing primary exposure in addition to phase 2 variables. For example, consider a study that aims to evaluate genetic variant on time to event response. One cannot readily obtain the genetic information from repository as much cost is required to validate the genetic information from the blood sample.
The proposed method aims to utilize information on all subjects in the estimating equa- tion, and therefore it seeks to fill in missing IPWs for non-subcohort controls by imputing missing phase 2 covariates. Intuitively, this method is valid as the subcohort is selected at random from the full cohort; the phase 2 variables are missing completely at random. Therefore, estimated values of partially missing covariates based on the random sample of the full cohort should not deviate too much from the true values if (1) the sampling was truly done in random fashion, and (2) imputation model is a correct model when parametric model is used, or nonparamteric estimation is used. Simulation results indicated that para- metric methods to estimate missing phase 2 variables can easily fail to improve efficiency when phase 2 variables are continuous.
The imputation method differs from the previously developed methods (in the standard associational context) which seek to utilize information available from the full cohort. In addition, we seek to make use of all subjects in the estimating estimation to improve effi- ciency in the case-cohort analysis. We do not require separate surrogate measurements of phase 2 variables as in Borgan et al. [2000]; Kulich and Lin [2004]; Breslow et al. [2009a,b], but time-varying confouders themselves can serve as surrogates (e.g., baseline CD4 or viral load can serve as surrogate of the following CD4 or viral load information).
The proposed estimator would be more efficient than estimators based on (4.4) or (4.5), because we use all subjects in the estimation step. Further, it could sometimes be more efficient than the full cohort estimator if imputed values are less variable than the true values (this is possible in some range of covariates). Nonetheless, bias would become bigger in such cases so MSE compared to the full cohort analysis would be larger.
Chapter 5
Summary and Future Research
In summary, we considered estimating the causal hazard ratios of MSCMs via inverse probability weighting in full cohort and the case-cohort studies. We established asymp- totic theories for estimators that maximize corresponding WPPLs under certain regularity conditions, via martingale and counting process formulation. In addition we proposed new variance estimators which could be more accurate than the robust variance estimators when sample size is small. Framing the problem using standard counting process and martingale theory readily enables practical implementation of the methods using existing survival anal- ysis software. However, implementing MSCM for the case-cohort design was shown to be not fully efficient. Therefore, we explored an imputation method that could lead to more efficient inference in the case-cohort MSCM analysis.
As we framed the problem of estimating the causal hazard ratios of MSCMs using counting processes and martingales, we may consider fitting MSCMs to data from nested case-control studies or in the presence of competing risks as next projects. Also, researchers have found that a main challenge of implementing MSMs in practice is difficulty in esti- mating inverse probability weights [Cole and Hern´an, 2008; Howe et al., 2011; Kang and Schafer, 2007; Lefebvre, Delaney and Platt, 2008; Mortimer et al., 2005]. It has been shown that results of using MSMs via inverse-probability-weighting could be highly sensitive to model misspecification of treatment assignment model, when even number of study visits is moderate. Therefore, doubly-robust-estimation of the causal hazard ratio of MSCMs in the presence of case-cohort sampling, or combining covariate balancing propensity score method proposed by Imai and Ratkovic [2014] in the inverse-probability-weighted estima- tion of MSCM hazard ratio could be a topic of future work.
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